Research On Discontinuous Galerkin Methods For Two-dimensional Interface Problems With Discontinuous Diffusion-coefficients

In this thesis, we mainly focus on the second-order interface problems with discontinuous diffusion-coefficients in two-dimensional domain, and study the partially penalty immersed finite element (PIFE) method based on Crouzeix-Raviart element on unfitted triangular meshes and the local discontinuous Galerkin (LDG) method on fitted triangular meshes. Meanwhile, to remain the property of the discontinuous Galerkin (DG) method that the computation can proceed element by element, and relax the strict time-step restriction (?t= O(hmin2)) nec-essary for explicit schemes, we study the implicit integration factor (IIF) method, which improves the computational efficiency greatly. The thesis is mainly divided into two parts.In the first part, we present the PIFE method on unfitted triangular grids for second-order elliptic interface problems. Firstly, on interface elements cut through by the interface curve, we construct the immersed finite element (IFE) spaces where the interface jump conditions are imposed, and study their proper-ties. Then, three PIFE schemes are given based on the symmetric, nonsymmetric and incomplete interior penalty discontinuous Galerkin (IPDG) formulation, re-spectively. The solvability of the method is proved and optimal error estimates in the energy norm are obtained. Finally, numerical experiments with piece-wise constant and definite-positive matrix diffusion-coefficients are presented to confirm the validity and optimal-order convergence accuracy of PIFE schemes.In the second part, on fitted triangular meshes, we apply the LDG method in-to homogeneous and non-homogeneous parabolic interface problems. The method can derive the numerical solutions and approximations for fluxes at the same time. This part is divided into two steps.The first step, we employ the LDG method to parabolic interface problems with homogeneous and non-homogeneous jump conditions, analyze the stability of the semi-discrete LDG scheme, and derive its a prior error estimate. With the help of explicit schemes, numerical experiments are conducted to verify the optimal and suboptimal convergence rate for the solutions and the fluxes, re-spectively, which is consistent with the theoretical analysis of the LDG method. Here, we observe that the LDG method for solving the non-homogeneous interface problems takes the homogeneous formulation with a special choice of numerical fluxes that incorporates the jump conditions across interface. Thus, not only the analysis but also the code can be easily obtained within the same framework.The second step, we find a more effective scheme for temporal discretiza-tion. Firstly, we apply the second-order IIF method coupled with LDG method (IIF-LDG) for solving the reaction-diffusion system. The method derives the numerical solutions and approximations for fluxes at the same time. In addition, it keeps the advantage of DG method that the computation can be conducted element by element, and allows us to use a larger time-step (At= O(hmin)), so that saves the computational time. Then we use the method to solve parabolic interface problems and give their full-discrete schemes. Two kinds of numeri-cal experiments with discontinuous constant and nonlinear diffusion-coefficients confirm the efficiency and convergence accuracy in space of the method. Fur-thermore, compared with the CPU time cost by the second-order explicit Runge-Kuttta time discretization, we observe that the IIF-LDG method indeed saves the computational time and improves the computational efficiency.